Geodesic
Uniqueness conditions and numerical approximation of the solution to M.M. Lavrentiev's integral equation
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 25 (2022) no. 4, pp. 441-458 
M. Yu. Kokurin; V. V. Klyuchev. Uniqueness conditions and numerical approximation of the solution to M.M. Lavrentiev's integral equation. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 25 (2022) no. 4, pp. 441-458. http://geodesic.mathdoc.fr/item/SJVM_2022_25_4_a8/
@article{SJVM_2022_25_4_a8,
     author = {M. Yu. Kokurin and V. V. Klyuchev},
     title = {Uniqueness conditions and numerical approximation of the solution to {M.M.~Lavrentiev's} integral equation},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
     pages = {441--458},
     year = {2022},
     volume = {25},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJVM_2022_25_4_a8/}
}
TY  - JOUR
AU  - M. Yu. Kokurin
AU  - V. V. Klyuchev
TI  - Uniqueness conditions and numerical approximation of the solution to M.M. Lavrentiev's integral equation
JO  - Sibirskij žurnal vyčislitelʹnoj matematiki
PY  - 2022
SP  - 441
EP  - 458
VL  - 25
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/SJVM_2022_25_4_a8/
LA  - ru
ID  - SJVM_2022_25_4_a8
ER  - 
%0 Journal Article
%A M. Yu. Kokurin
%A V. V. Klyuchev
%T Uniqueness conditions and numerical approximation of the solution to M.M. Lavrentiev's integral equation
%J Sibirskij žurnal vyčislitelʹnoj matematiki
%D 2022
%P 441-458
%V 25
%N 4
%U http://geodesic.mathdoc.fr/item/SJVM_2022_25_4_a8/
%G ru
%F SJVM_2022_25_4_a8

Voir la notice de l'article provenant de la source Math-Net.Ru

M.M. Lavrentiev's linear integral equation arises as a result of a special transformation of a nonlinear coefficient inverse wave sensing problem. The completeness of the set of products of regular harmonic functions and Newtonian potentials supported by a segment is proved. As a corollary, we establish the uniqueness of the solution to M.M. Lavrentiev's equation and a related inverse problem of wave sensing. We present results of an approximate solution of this equation by using parallelization of calculations.

[1] M. M. Lavrentev, “Ob odnoi obratnoi zadache dlya volnovogo uravneniya”, Dokl. AN SSSR, 157:3 (1964), 520–521

[2] M. M. Lavrentev, “Ob odnom klasse obratnykh zadach dlya differentsialnykh uravnenii”, Dokl. AN SSSR, 160:1 (1965), 32–35

[3] A. B. Bakushinskii, A. I. Kozlov, M. Yu. Kokurin, “Ob odnoi obratnoi zadache dlya trekhmernogo volnovogo uravneniya”, Zhurn. vychisl. matem. i mat. fiziki, 47:3 (2003), 1201–1209

[4] B. R. Vainberg, Asimptoticheskie metody v uravneniyakh matematicheskoi fiziki, Izd-vo MGU, M., 1982

[5] V. G. Romanov, “O gladkosti fundamentalnogo resheniya dlya giperbolicheskogo uravneniya vtorogo poryadka”, Sib. mat. zhurnal, 50:4 (2009), 883–889

[6] A. I. Kozlov, M. Yu. Kokurin, “Ob integralnykh uravneniyakh tipa M.M. Lavrenteva v koeffitsientnykh obratnykh zadachakh dlya volnovykh uravnenii”, Zhurn. vychisl. matem. i mat. fiziki, 61:9 (2021), 1492–1507

[7] M. Klibanov, J. Li, W. Zhang, “Linear Lavrent'ev integral equation for the numerical solution of a nonlinear coefficient inverse problem”, SIAM J. Appl. Math., 81:5 (2021), 1954–1978

[8] M. M. Lavrentev, V. G. Romanov, S. P. Shishatskii, Nekorrektnye zadachi matematicheskoi fiziki i analiza, Nauka, M., 1980

[9] A. G. Ramm, Mnogomernye obratnye zadachi rasseyaniya, Mir, M., 1994

[10] A. L. Bukhgeim, G. V. Dyatlov, V. B. Kardakov, E. V. Tantserev, “Edinstvennost v odnoi obratnoi zadache dlya sistemy uravnenii uprugosti”, Sib. mat. zhurnal, 45:4 (2004), 747–757

[11] M. Yu. Kokurin, S. K. Paimerov, “Ob obratnoi koeffitsientnoi zadache dlya volnovogo uravneniya v ogranichennoi oblasti”, Zhurn. vychisl. matem. i mat. fiziki, 48:1 (2008), 117–128

[12] M. Yu. Kokurin, “On a multidimensional integral equation with data supported by low-dimensional analytic manifolds”, J. of Inverse and Ill-Posed Probl, 21:1 (2013), 125–140

[13] M. Yu. Kokurin, “O polnote proizvedenii garmonicheskikh funktsii i edinstvennosti resheniya obratnoi zadachi akusticheskogo zondirovaniya”, Mat. zametki, 104:5 (2018), 708–716

[14] M. Yu. Kokurin, “O polnote proizvedenii reshenii uravneniya Gelmgoltsa”, Izv. vuzov. Matematika, 2020, no. 6, 30–35

[15] M. Yu. Kokurin, “Polnota asimmetrichnykh proizvedenii reshenii ellipticheskogo uravneniya vtorogo poryadka i edinstvennost resheniya obratnoi zadachi dlya volnovogo uravneniya”, Diff. uravneniya, 57:2 (2021), 255–264

[16] A. B. Bakushinskii, A. S. Leonov, “Ekonomichnyi chislennyi metod resheniya koeffitsientnoi obratnoi zadachi dlya volnovogo uravneniya v trekhmernom prostranstve”, Zhurn. vychisl. matem. i mat. fiziki, 58:4 (2018), 561–574

[17] M. V. Klibanov, V. G. Romanov, “Reconstruction procedures for two inverse scattering problem without the phase information”, SIAM J. Appl. Math., 76:1 (2016), 178–196

[18] V. G. Romanov, “Zadacha ob opredelenii koeffitsienta dielektricheskoi pronitsaemosti po modulyu rasseyannogo elektromagnitnogo polya”, Sib. mat. zhurnal, 58:4 (2017), 916–924

[19] V. G. Romanov, “Obratnye zadachi bez fazovoi informatsii, ispolzuyuschie interferentsiyu voln”, Sib. mat. zhurnal, 59:3 (2018), 626–638

[20] V. A. Dedok, A. L. Karchevskii, V. G. Romanov, “Chislennyi metod opredeleniya dielektricheskoi pronitsaemosti po modulyu vektora elektricheskoi napryazhennosti elektromagnitnogo polya”, Sib. zhurn. industr. matematiki, 22:3 (2019), 48–58

[21] L. D. Kudryavtsev, Kurs matematicheskogo analiza, v. 2, Drofa, M., 2004

[22] V. A. Ditkin, A. P. Prudnikov, Integralnye preobrazovaniya i operatsionnoe ischislenie, GIFML, M., 1961

[23] M. A. Lavrentev, B. V. Shabat, Metody teorii funktsii kompleksnogo peremennogo, Nauka, M., 1965

[24] I. Stein, G. Veis, Vvedenie v garmonicheskii analiz na evklidovykh prostranstvakh, Mir, M., 1974

[25] V. V. Voevodin, Yu. A. Kuznetsov, Matritsy i vychisleniya, Nauka, M., 1984